University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2013 An Automated Finite Element Analysis Framework for the Probabilistic Evaluation of Composite Lamina Properties Jonathan Phillips Weigand University of Tennessee - Knoxville, jweigan2@utk.edu Recommended Citation Weigand, Jonathan Phillips, "An Automated Finite Element Analysis Framework for the Probabilistic Evaluation of Composite Lamina Properties. " Master's Thesis, University of Tennessee, 2013. http://trace.tennessee.edu/utk_gradthes/2649 This Thesis is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu.
To the Graduate Council: I am submitting herewith a thesis written by Jonathan Phillips Weigand entitled "An Automated Finite Element Analysis Framework for the Probabilistic Evaluation of Composite Lamina Properties." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Civil Engineering. We have read this thesis and recommend its acceptance: Edwin G. Burdette, Richard Bennett (Original signatures are on file with official student records.) Stephanie C. TerMaath, Major Professor Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School
An Automated Finite Element Analysis Framework for the Probabilistic Evaluation of Composite Lamina Properties A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Jonathan Phillips Weigand December 2013
ii Copyright 2013 by Jonathan Phillips Weigand All rights reserved.
iii Acknowledgements I would like to express a deep appreciation for my major professor, Dr. Stephanie TerMaath, who has provided me a remarkable opportunity and foundation to perform exciting and innovative research. She has displayed her unwavering passion and excitement in regard to teaching and research and has been a great personal and academic mentor. I would also like to thank my committee members, Dr. Edwin Burdette and Dr. Richard Bennett, who have helped to shape my intellectual aspirations and have shown a relentless commitment to excellence. Without the guidance of these great people, this thesis would not have been possible.
iv Abstract This thesis outlines the development of computational modeling tools used to predict the elastic properties of composite lamina from representative volume elements (RVE) using numerical methods. The homogenization approach involves the use of Gauss s Theorem to simply the average volumetric strain integral into a surface integral containing which is defined by surface displacements and their direction. Simulations of RVEs under specific loading conditions (longitudinal tension or shear and transverse tension or shear) are then performed in the software package ABAQUS to obtain the surface displacements. It was found that obtaining quality meshes and applying periodic boundary conditions for each RVE was very user and time-intensive, thus, modeling tools were developed to automate the modeling process. The homogenized composite lamina properties are obtained using C++ code to generate models, batch scripts to run successive simulations in ABAQUS, and C++ code to extract surface displacements from the output data and calculate the final properties. The model generation code contains many user-controllable features such as constituent material properties, fiber volume fraction, mesh density, and type of applied boundary conditions. The modeling tools are then applied in a sensitivity study modeling a unit cell to identify which constituent properties have the largest impact on out-of-plane lamina properties. The framework is then extended to model lamina RVEs consisting of two different fiber materials. The location and percent composition of the two different fiber materials are varied to analyze the effect on out-of-plane lamina properties and identify optimal material combination.
v Table of Contents Chapter 1 Introduction... 1 Motivation... 1 Research Objective... 1 Thesis Outline... 4 Chapter 2 Composite Material Overview and Literature Review... 5 Composite Material Introduction... 5 Applications... 6 Aerospace... 6 Aircraft... 6 Other Applications... 7 Analysis Methods Overview... 7 Micromechanical Approach... 8 Boundary Element Method... 10 Overview... 10 Implementations... 11 Finite Element Method... 12 Overview... 12 Implementations... 12 Method Comparison... 14 Motivation for an Automated Approach... 14
vi Chapter 3 Sensitivity Analysis of Out-of-Plane Composite Properties... 16 Introduction... 16 Model Structure... 17 Automation Framework... 17 Finite Element Model... 20 Homogenization Approach... 20 Longitudinal and Transverse Tension Loading... 23 Transverse Shear Loading... 25 Validation of Predicted Properties... 27 Investigation of Poisson s Effect... 30 Single Variant Sensitivity Analysis... 31 Core Methodology... 31 Determination of Input Parameter Range... 32 Results and Discussion... 33 Conclusion... 40 Chapter 4 Sensitivity Analysis of Composite Lamina with Multiple Fiber Materials... 41 Introduction... 41 Finite Element Model... 43 Model Structure... 43 Validation... 43 Sensitivity Analysis of Composite Lamina Consisting of Multiple Fiber Types... 45 Fiber Percentage Study... 46
vii Fiber Configuration Study... 50 Conclusion... 51 Chapter 5 Conclusion... 52 List of References... 53 Vita... 58
viii List of Tables Table 1. Summary of Convergence Study 19 Table 2. Mechanical Properties of Fiber and Matrix Materials 27 Table 3. Mechanical Properties of Carbon/Epoxy Composite Lamina (Fv=0.6) 29 Table 4. Comparison of Lamina Properties Obtained from FEA and Self-Consistent Field Method 29 Table 5. Comparison of Lamina Properties Relative to Boundary Conditions (GPa) 31 Table 6. 3501-6 Epoxy Resin Properties from Literature Search 34 Table 7. AS4 Carbon Fiber Properties from Literature Search 34 Table 8. Comparison of Effective Stress and Properties Obtained from the Lamina and Unit Cell Models (AS4 Carbon / 3501-6 Epoxy) 45 Table 9. Avg. Properties of AS4 Carbon and Kevlar-49 from Literature Search (GPa) 46
ix List of Figures Figure 1. Vision of the Probabilistic Approach to Simulation... 2 Figure 2. Multi-Scale Composite Framework... 3 Figure 3. Composite Reinforcement Regimes... 5 Figure 4. Unit Cell Compared to RVE of Lamina... 9 Figure 5. Typical Unit Cells for FRCs... 9 Figure 6. Comparative Example of Discretization Methods... 10 Figure 7. Comparison of Deformed Shape Assumptions... 13 Figure 8. Framework for Sensitivity Analysis and Design Optimization... 17 Figure 9. Example Meshes Generated for Convergence Study... 19 Figure 10. Depiction of the Unit Cell and Coordinate System... 21 Figure 11. Normal Loads and Displacements for FEA Unit Cell Model... 23 Figure 12. Compatible Deformation of the Unit Cell... 24 Figure 13. Transverse Shear Load and Displacements for FEA Unit Cell Model... 26 Figure 14. Comparison of Deformed Shape with and without Poisson s Effect... 30 Figre 15. Comparison of von Mises Stress with Varying Boundary Conditions... 31 Figure 16. Sensitivity of E 2L... 35 Figure 17. Sensitivity of v 23L... 36 Figure 18. Sensitivity of G 23L... 37 Figure 19. Sensitivity of E 2L Relative to Constituent Properties... 39 Figure 20. Sensitivity of v 23L Relative to Constituent Properties... 39
x Figure 21. Sensitivity of G 23L Relative to Constituent Properties... 39 Figure 22. Example RVE of a Lamina... 42 Figure 23. Comparison of Deformed Shape and Effective von Mises Stress Contours Obtained from the Lamina and Unit Cell Models... 44 Figure 24. Lamina Combinations for Sensitivity Analysis (Black: Kevlar-49)... 47 Figure 25. Effect of Increasing Kevlar-49 Percentage on E 2L and v 23L... 49 Figure 26. Effect of Increasing Kevlar-49 Percentage on G 23L... 49 Figure 27. Bi-Fiber Lamina Optimization Problem... 49 Figure 28. Lamina Configurations for Sensitivity Analysis (Black: Kevlar-49)... 50 Figure 29. Effect of Configuration on E 2L and v 23L... 51
xi List of Abbreviations FRC FE FEA RVE OFAT GUI CSV fiber reinforced composite finite element finite element analysis representative volume element one-factor-at-a-time graphical user interface comma-separated values
xii List of Symbols E m v m E 1f E 2f E 3f v 12f v 23f G 12f G 23f E 1L E 2L E 3L v 12L v 23L G 12L G 23L V F U ij ij ij ij W l m n u 1 u 2 u 3 P 2 T 23 x s S 1 S 2 elastic modulus of the matrix material Poisson s ratio of the matrix material elastic modulus in the x direction of the fiber material elastic modulus in the y direction of the fiber material elastic modulus in the z direction of the fiber material in-plane Poisson s ratio of the fiber material out-of-plane Poisson s ratio of the fiber material in-plane shear modulus of the fiber material out-of-plane shear modulus of the fiber material elastic modulus in the x direction of the lamina elastic modulus in the y direction of the lamina elastic modulus in the z direction of the lamina in-plane Poisson s ratio of the lamina material out-of-plane Poisson s ratio of the lamina material in-plane shear modulus of the lamina material out-of-plane shear modulus of the lamina material fiber volume fraction strain energy volumetric average stress tensor volumetric average strain tensor stress tensor strain tensor work performed on the system unit cell dimension in the x direction unit cell dimension in the y direction unit cell dimension in the z direction boundary displacement in the x direction obtained from ABAQUS boundary displacement in the y direction obtained from ABAQUS boundary displacement in the z direction obtained from ABAQUS normal traction stress applied in the y direction shear traction stress traction applied in the y-z plane sample mean sample standard deviation surface at the fiber-matrix interface of the unit cell exterior surface of the unit cell
1 Chapter 1 Introduction Motivation To design optimized and reliable parts manufactured with advanced composite materials customized for particular applications, designers need the capability to efficiently asses the performance of engineered materials and customized composite parts subject to variable loading conditions. Currently, variability in constitutive properties and manufacturing outcomes result in large safety factors for composite parts. A probabilistic based simulation approach that quantifies material and manufacturing uncertainties through multiple scale levels could allow for engineers to more realistically predict composite part performance, develop design guidelines, and limit the costs associated with overdesign and extensive destructive testing programs (Figure 1). A progressive simulation environment would promote recognition of experiments needed to fill in knowledge gaps and provide validation data. Additionally, a fully automated probabilistic-based modeling framework could allow engineers to efficiently analyze many material combinations and configurations using preexisting constituent data. Research Objective Fiber reinforced composite (FRC) materials provide customizable structural properties, but predicting and evaluating the response of these materials are difficult due to the nearly unlimited material combinations and configurations used to make composite parts. Consequently, characterizing FRCs through repetitive experimental testing
2 Figure 1. Vision of the Probabilistic Approach to Simulation programs has proven to be very time-consuming and expensive posing a huge challenge to the industries using composite materials. The most favorable alternative is to characterize effective composite properties using numerical simulation methods that can continuously be refined and built upon to meet the needs of the industry. Therefore, this research is based on a probabilistic multi-scale framework that aims to develop computational tools that allow for efficient design of optimized and reliable composite parts. The framework consists of modeling fiber aligned materials across two scale levels (Figure 2) while incorporating the mechanical behavior and respective uncertainty at each scale level to obtain the effective engineering properties of the final composite material.
3 (a) unit cell (b) lamina Figure 2. Multi-Scale Composite Framework This thesis demonstrates an automated numerical approach for rapidly assessing unit cell level and lamina level properties within the proposed multi-scale framework. The unit cell consists of a single fiber surrounded by its respective matrix material while the lamina consists of a periodic combination of unit cells. The unit cell approach involves the assumption that the composite cross-section is perfectly periodic, which simplifies the mathematics for analytical formulations and provides for a computationally efficient method for predicting lamina properties [1]. The work presented in this thesis will focus on the development, validation, and application of an automated numerical approach that predicts the effective engineering properties of a composite lamina while capturing the influence of uncertainties in material constitutive properties, lamina configuration, and periodic boundary conditions. The numerical approach consists of user-friendly and readily-expandable computer programs written to rapidly generate a finite element model of a unit cell or representative volume element for a lamina, simulate the response to various loading conditions, and extract the data necessary to calculate homogenized lamina properties.
4 Thesis Outline An introduction to fiber reinforced composite materials and composite analysis methods will be discussed in chapter 2 of this thesis. Chapter 3 demonstrates the use of the developed modeling tools to perform an automated sensitivity analysis at the unit cell level to evaluate the influence of constituent properties and geometry on out-of plane lamina properties. The modeling tools consist of code written in C++ programming language to generate input files, automate the simulation process in the finite element code ABAQUS, extract boundary displacements, and calculate homogenized properties. The input file generation code has user-controllable features that allow for different constituent materials, load combinations, and boundary conditions. The results are validated according to analytical and experimental results found in the literature. Chapter 4 entails an extension of the computational tools in which two different fiber materials are introduced in a lamina representative volume element in order to examine the influence on out-of-plane lamina properties and identify the precise combination of fiber types to achieve optimized homogenized properties.
5 Chapter 2 Composite Material Overview and Literature Review Composite Material Introduction Military and industrial applications for fiber reinforced composite (FRC) materials have grown exponentially over the past five decades due to their desirable material properties. These properties include favorable strength to weight ratio, stiffness, and corrosion resistance compared to conventional structural materials such as steel and concrete. FRC materials consist of strong fiber strands embedded in a matrix material that acts to bind fibers together and provide multi-directional strength. Typically, the fibers are placed in the matrix material as continuous fiber strands stacked in multidirectional layers or short randomly distributed fibers (Figure 3). (a) fiber aligned composite (b) randomly oriented short fiber composite Figure 3. Composite Reinforcement Regimes With today s manufacturing capabilities, FRCs can be tailored to handle extreme loading and environmental conditions, reduce stress concentrations that arise from bolted
6 or welded connections, or provide unique performance advantages when combined with other materials such as stainless steel [2]. As a result, FRCs have applications across many disciplines, and they are outlined as follows. Applications Aerospace Structures sent into space such as antennae, optical instrumentation, and mirrors consist of lightweight yet exceptionally stiff graphite composites. These graphite composites are designed to have very low thermal and hygric expansion coefficients which allow them to have a high degree of multi-directional stability in extreme environments [3]. Aircraft Since the late 1950 s, fibrous reinforced materials have been increasingly used in military aircraft because they exhibit high strength-to-weight ratio, low electrical conductivity, high corrosion resistance, and stealth characteristics. Composite materials are increasingly being used for commercial airliners as well. Boeing Aerospace Company is ushering in a new era of commercial aircraft transportation with its new 787 Dreamliner which has an 8,000 mile range and the ability to transport 210-250 passengers in a single flight. The aircraft structure is designed to be roughly 50 percent by weight of carbon-epoxy and graphite-titanium composite, which will provide 20 percent less fuel burn, 30 percent less maintenance cost, and increased reliability compared to similar jets.
7 Other Applications Composite material applications also include automotive, marine, energy, infrastructure, armor, biomedical, and recreational sporting. An early application in the automotive industry occurred when GM replaced the steel rear leaf spring in its Corvette with a glass-epoxy spring to reduce the weight [3]. Carbon fiber composite is being used as a strong, lightweight material for newly developed medical and sports equipment. For example, prosthetic limbs are now being manufactured from carbon fiber to make movement more feasible and realistic. Even cycling shoes are being made of a carbon fiber because the material can be heated and shaped specifically for each rider s feet. The applications of FRC composite materials are forever growing and are only limited by the ability to accurately predict structural performance during the design process and efficiently manufacture unique parts. Analysis Methods Overview Various methods have been used to predict the effective properties of composite materials, and these methods are broadly categorized as analytical, numerical, or experimental approaches [3]. Theoretical methods provide closed-form expressions allowing for instantaneous homogenized property calculations [1, 4-10]. However, theoretical methods often include gross material or geometric simplifications that result in over-simplified material property prediction particularly in the transverse direction (E 23L, v 23L, G 23L ). Experimental methods are necessary to validate the relation between constituent properties and average lamina properties, but as mentioned before, testing regimes are labor-intensive, time-consuming, and expensive. Numerical approaches have
8 been shown to provide the most accurate property predictions, but have been considered to be too time-consuming for practical use until recently. Therefore, numerical methods in combination with today s available computing resources provide a favorable alternative to composite material modeling, which is why they were chosen for the focus of this study. Micromechanical Approach Having the means to accurately and efficiently predict the effective properties of heterogeneous media is critical to tailoring advanced composite materials. In classical homogenization theory, multi-phase materials such as carbon-epoxy are described in terms of unit cells or representative volume elements (RVE) that imitate the composite microstructure [11]. A unit cell consists of a single fiber and the surrounding matrix material while a lamina RVE consists of multiple fibers surrounded by matrix material. In the literature, the terms unit cell and RVE are sometimes used interchangeably, but Figure 4 displays how each will be defined in this thesis. The unit cell and RVE are sized relative to its constituents to allow for the average properties to remain independent of location within the element. Figure 5 depicts the unit cells most typically used for fibrous composites. Hashin provided an extensive review on periodic RVEs and summarized the concept therein [1]. In theory, boundary conditions can be enforced on a unit cell or RVE such that it is forced to behave as part of larger composite lamina [12-16]. The simplified unit cell or RVE allows for the constituent materials and effective engineering property relation to be efficiently studied through numerical methods.
9 (a) unit cell (b) RVE of lamina Figure 4. Unit Cell Compared to RVE of Lamina (a) square packing unit cell (b) hexagonal packing unit cell Figure 5. Typical Unit Cells for FRCs
10 Boundary Element Method Overview The Boundary Element Method (BEM) is a computational method for approximating the solution to partial differential equations posed as equivalent integral equations known as boundary integrals [17]. For a given boundary value problem, boundary conditions are introduced as boundary values in the boundary integrals rather than values throughout the problem domain of the partial differential equation. The boundary integrals can be used to obtain the solution at any point within the problem domain. BEM involves only surface discretization which can render BEM more computationally efficient than volume discretization methods such as the Finite Element Method (Figure 6). For this reason, BEM has been used by researchers to study the material behavior of periodic composite materials. However, BEM was not the method chosen for the focus of this thesis and the reasons why will be discussed at the end of this chapter. (a) Boundary Element Method (b) Finite Element Method Figure 6. Comparative Example of Discretization Methods
11 Implementations The BEM can provide a straightforward modeling approach when dealing with materials of repetitive microstructure such as FRC composites because of the surface meshing advantages. Kaminski used a BEM based approach to model a RVE of linearly elastic, periodic heterogeneous composite media [18, 19]. The effective elastic constants are computed using strain energy equivalency between the actual configuration and the idealized homogenous medium. The calculated properties were shown to be consistently higher than the properties obtained using FEM-based formulations, and the difference was attributed to the computational procedure used to spatially average the stress tensor components within the RVE. Okada et al. presented a BEM-based homogenization technique using the method of weighted residuals to derive the integral equations and ultimately the homogenization formulation [20]. The approach involves the assumption that the fiber volume fraction must remain the same across multiple until cells within the local periodic microstructure and that the material constituents must be isotropic. Chen presented a BEM based technique to study the effects of a third interphase layer located between the fiber and matrix materials [21]. However, the BEM solver was found to run much slower than the commercial FEM software, ANSYS, in which the same analysis was performed.
12 Finite Element Method Overview Like BEM, the Finite Element Method (FEM) is a numerical approach for approximating the solutions to partial differential equations for boundary value problems. Simple shape functions within many discrete subdomains (finite elements) are used to approximate the behavior of a continuous medium. These elemental shape functions allow for field quantity values, such as displacement, to be determined at any location within each element. FEM involves volume discretization rather than surface discretization and provides a natural approach for structural analysis problems. FEM is widely used in the composite modeling community because of its versatility and ability to capture local behavioral effects [22]. Implementations Blacketter and King utilized a two-dimensional, quarter-symmetric FE model of a RVE to study the effect of matrix and interface properties on the transverse tensile and shear strengths of fiber reinforced materials [23, 24]. Garnich and Karami developed a three dimensional FE model to study the effects of fiber waviness on the structural stiffness and strength properties of fiber reinforced composites [25, 26]. Medeiros modeled multiple RVEs to study the influence of fiber arrangement and shape on homogenized properties [27]. Naik performed a similar analysis on fiber arrangement, but additionally studied bi-directional packing sequences [28].
13 The literature revealed inconsistencies regarding the correct application of boundary constraints to force a unit cell or RVE to behave as part of a larger composite lamina. Naik, Brockenbrough, and Caruso [29-31] made the assumption that the deformed shape remained plane under transverse shear loading, while Adams and Crane and Sun and Vaidya [32, 33] assumed the deformed shaped did not remain plane (Figure 7). Some researchers have chosen to neglect Poisson s effect due to the notion that boundaries without a primary normal load remain stationary as part of the larger lamina. Many FE modeling approaches have been developed over the past two decades [34-40], yet there is still disagreement in the literature regarding the correct application of boundary displacement constraints. (a) Deformation remains in plane (b) Deformation occurs out of plane Figure 7. Comparison of Deformed Shape Assumptions
14 Method Comparison While BEM may be computationally advantageous in some cases, its applicability is mostly limited to linear problems with simple microstructure [41]. BEM is ill-suited for problems involving nonlinearities or complex micro-geometry such as matrix voids or micro-cracking because each inclusion must discretized separately. The surface discretization scheme creates non-local connectivity in elements rendering fully populated matrices which, as a result, causes computational time to grow exponentially with problem complexity and size. Meanwhile, FEM provides a natural approach for analyzing complex structures across all scales, and because FEM matrices are often banded, computation time grows linearly with problem complexity and size [22]. FEM exhibits superior promise for handling complex micro-geometry such as matrix voids and micro-cracking because discretization of the actual geometry enforces local connectivity of elements. Additionally, FEM has been shown to provide accurate solutions [33, 42] when compared to analytical [1, 10, 43] and experimental [44, 45] results and is available in many commercial packages. Motivation for an Automated Approach Many of the models presented in the literature are formulated in two dimensions, involve axis symmetry, and involve conflicting assumptions on the boundary conditions applied to the RVE. The literature has also revealed that computation time, model efficiency, and usability is rarely a focus in the aforementioned numerical approaches, and it has been found that generating quality meshes and applying periodic boundary conditions for RVEs using a FEA software package such as ABAQUS can be a very
15 meticulous and time-consuming process. Therefore, providing modeling tools that automate the mesh generation process and give the user the ability to control the nature of the applied boundary conditions (e.g. include or neglect Poisson s effect) is a primary task in the development of an automated, probabilistic modeling framework.
16 Chapter 3 Sensitivity Analysis of Out-of-Plane Composite Properties Introduction The futuristic vision for aircraft analysis is a simulation environment where engineers can create and explore unlimited configurations to develop reliable and optimized parts. Currently, the design of customized parts is accomplished through expensive and time-consuming comprehensive test programs. In addition to streamlining material design, an encompassing simulation environment will promote effective design of experiments that will limit material testing, fill in knowledge gaps, and provide quality validation data. With today s computing resources, an automated, probabilistic-based multi-scale numerical modeling framework will allow for composite designers to economically engineer materials, explore material configurations, and optimize part performance. This chapter outlines the foundation for the framework to achieve the aforementioned vision. Newly developed automated modeling tools are used to perform multi-scale sensitivity analyses and design optimization on proposed material designs through the use of finite element modeling (Figure 8). The computational tools provide the ability to run multiple FEAs, extract displacements from each simulation, and calculate the effective lamina properties with a single initiation of the program. Finite element approaches have been considered user-intensive and computationally demanding. However, the developed computational tool automates the FE modeling
17 process which allows the user to quickly identify constituent parameters that have the most influence on lamina level properties. To demonstrate the automated modeling tool, a sensitivity analysis is performed to evaluate the influence of constituent properties on effective composite lamina properties. The modeling framework is then extended to investigate composite lamina consisting of variable ratios of two different fiber materials. Figure 8. Framework for Sensitivity Analysis and Design Optimization Model Structure Automation Framework Automation of the FE modeling process consists of three main programming tasks: pre-processing to build each customized unit cell or lamina model, scripting to drive successive simulations, and post-processing to extract all necessary data. The preprocessing stage is performed through code written in C++ programming language, which, upon compilation, rapidly generates input files from the following user specified
18 parameters: degree of mesh density, matrix (E m and v m ) and fiber (E 1f, E 2f, v 12f, v 23f, and G 12f ) material properties, fiber volume fraction (F v ), load type (longitudinal or transverse tension and shear) and nature of applied boundary conditions (include or neglect Poisson s effect). The major tasks performed by the code consist of generating a quality mesh for each unit cell, defining contact and load surfaces, and defining boundary displacement constraints that enforce periodic behavior in the unit cell or lamina RVE. The meshing algorithm ensures quality element aspect ratios even as the fiber volume fraction exceeds 0.5, which was observed to be very difficult using the graphical user interface (GUI) in ABAQUS. Quality shaped elements located along the fiber-matrix interface will allow for the bond state to be examined in future studies. The automated definition of boundary conditions eliminates the time and possibility of error associated with this essential task. The chosen displacement constraints are discussed in detail in the following section of this paper. The final portion of the program generates a batch file which contains the scripting necessary to run successive simulations in ABAQUS Standard and create separate comma-separated values (CSV) files containing the surface displacements from each run. The C++ code and batch scripting bypass the need to use the ABAQUS GUI which enables convergence studies to be performed quickly and with minimal user intervention (Figure 9).
19 (a) 3,208 elements (b) 6,920 elements (c) 29,368 elements (d) 80,472 elements Figure 9. Example Meshes Generated for Convergence Study Table 1 contains a summary of a mesh convergence study. The displacement terms (u 1, u 2, and u 3 ) are the boundary displacements in the principal directions obtained from longitudinal tension, transverse tension, and transverse shear load conditions. The purpose for these load conditions and displacements will be discussed later in this chapter. It was determined that mesh c provided both suitable displacement convergence and computational efficiency. Mesh Mesh Density Details Elements Table 1. Summary of Convergence Study Avg. Run (sec) Longitudinal Tension u 1 (mm) u 2 = u 3 (mm) Transverse Tension u 2 (mm) u 3 (mm) Transverse Shear u 2 = u 3 (mm) a 3,208 3 0.0483-0.0122 0.7151-0.2501 1.1166 b 6,920 6 0.0482-0.0122 0.7150-0.2503 1.1142 c 29,368 20 0.0481-0.0121 0.7145-0.2502 1.1111 d 80,472 120 0.0481-0.0121 0.7143-0.2502 1.1099
20 The final portion of the modeling tool is a C++ program that retrieves the boundary displacements and runtime from each simulation, calculates the effective (homogenized) elastic lamina properties, and outputs the lamina properties, constituent properties, and runtime into a single CSV file. This comma-separated value file displays the all results in column fashion in spreadsheet management software (Microsoft Excel) such that the correlation between the constituent properties and lamina properties can be readily examined. Ultimately, the modeling tools permit users to obtain all six independent elastic lamina properties (E 1L, E 2L, v 12L, v 23L, G 12L, and G 23L ) in a single initiation of the program. Finite Element Model Homogenization Approach The distribution of fibers within an actual composite lamina is random through the cross-section. However, most micromechanical models simplify the micro-geometry by assuming a periodic fiber arrangement allowing for a single unit cell or RVE to characterize the lamina. The unit cell for this analysis consists of a single fiber surrounded by its respective matrix material indicative of a square packing sequence (Figure 10).
21 Figure 10. Depiction of the Unit Cell and Coordinate System Examining the behavior of the unit cell subject to longitudinal (in-plane) and transverse (out-of-plane) loading schemes can give insight into the lamina level behavior. Because the elastic properties of the fiber and matrix material are usually quite different, the stress and strain fields resulting from uniform loading are not uniform as in a homogenous medium. Therefore, special measures must be taken in order to confirm that simple stress and strain relations may be used to calculate the average elastic properties of the two combined materials. Sun and Vaidya [33] formulated a homogenization approach by proving that the strain energy stored homogenous unit cell (eq. 1-3) is equal to that in a heterogeneous unit cell provided that perfect bonding occurs at the fiber-matrix interface (eq.1). U 1 dv 2 v ij ij (1) 1 ij V V ( ij x,y,z ) dv (2) 1 ij V ij ( x,y,z ) dv (3) V
22 Because all unit cells are assumed to be identical across the composite section, the stress and strain gradients resulting from uniform loading conditions are periodic. The average volumetric strain within the unit cell is related to the boundary displacements through Gauss s Theorem, which relates the flow of a vector field through a surface to the behavior of the internal vector field. Gauss s Theorem allows for the average volumetric strain in the heterogeneous unit cell to be converted to surface integrals that occur at the outer fiber surface (S 1 ), inner matrix surface (S 1 ), and outer matrix surface (S 2 ). The surface integrals are characterized by the displacement at the surface and the unit normal (direction) of the displacement. The two surface integrals along the fiber matrix interface are equal and opposite, which results in a final surface integral that contains only boundary displacements and the associated unit normals (eq.4). 1 1 1 1 ij (u i j j i S in j u jni )ds (u n u n )ds V 2 2 2 1 2V 2 S1 S1 S 2 u n u n ds i j j i (u n u n i j j i ) ds (4) Because surface displacement constraints are enforced on the unit cell such that all surface displacements are known, only these boundary displacements are needed to calculate the average volumetric strain ( ij ). This approach is favorable because obtaining the surface displacements is much less rigorous than attaining the strain in every element. This average volumetric strain can then be used to determine the homogenized (averaged) properties.
23 Longitudinal and Transverse Tension Loading A longitudinal tension load is modeled by a uniform load (P 1 ) acting on the front and back faces (x = + l) of the unit cell, while a transverse tension load is modeled by a uniform load acting on the right and left faces (y = + m) of the unit cell (Figure 11). The boundary displacement constraints are specified in the FE model as such, u (l,y,z ) 1 1 u ( l,y,z ) u2 ( x,m,z ) u2( x, m,z ) (5) u ( x,y,n ) 3 3 u ( x,y, n ) where u 1, u 2, and u 3 are the surface displacements in the x, y, and z directions computed with FEA. These boundary conditions allow contraction due to Poisson s effect. (a) Longitudinal Tension (b) Transverse Tension Figure 11. Normal Loads and Displacements for FEA Unit Cell Model The surfaces are assumed to remain plane during normal loading in order to force compatible behavior of the unit cell (Figure 10).
24 (a) Compatible deformation (b) Incompatible deformation Figure 12. Compatible Deformation of the Unit Cell Continuing to follow Sun and Vaidya [33] for the longitudinal tension load case, the average longitudinal is calculated using the definition of strain. u 1 11 l (6) The internal strain energy stored in the unit cell is equated to: 1 U 11 11 V (7) 2 The external work performed on the unit cell by the uniform load is formulated by: W 1 P1 2m 2n 2u1 (8) 2 Equating the two according to the work-energy principal results in: 1 1 11 11 V P1 2m 2n 2u1 (9) 2 2 By substituting eq. (6) into eq. (9), the average stress in the longitudinal direction is proven to equal the applied uniform tensile load: 11 P 1 (10)
25 Thus, the effective longitudinal modulus and Poisson s ratio are calculated as E1 L P1 l and u 1 v12 L u2 (11) u 1 because the unit cell side lengths are identical. In a similar fashion, the transverse modulus and Poisson s ratio are determined by: E2 L P2 m and u 2 u3 v23 L (12) u 2 Transverse Shear Loading A transverse shear load is modeled by counter-clockwise surface tractions on the right and left surfaces (y = + m) and clockwise surface tractions on the top and bottom surfaces (z = + n) of the unit cell (Figure 10). Sun and Vaidya analyzed a 3 by 3 cell model subject to transverse shear in order to determine the correct boundary displacement constraints to place on the RVE [33]. The middle cell is typically used as the representative cell because it is adequately distanced from the free edges. The analysis results revealed that the surfaces of the middle cell do not remain plane as assumed in previous modeling approaches [29-31] and that only periodicity and symmetry conditions need to be satisfied.
26 Figure 13. Transverse Shear Load and Displacements for FEA Unit Cell Model Therefore, the boundary displacement constraints applied to the unit cell are determined as such: u (l,y,z ) u ( 1,y,z ) 1 1 u3 ( x,m,z ) u3( x, m,z ) u 2 ( x,m,z ) u 2 ( x, m,z ) (13) u ( x,y,n ) 2 u2( x,y, n ) u 3 ( x,y,n ) u 3 ( x,y, n ) Sun and Vaidya also determined that lateral straining does not occur at the boundaries of the middle cell [33]. Therefore, the RVE surface displacements in the directions of the applied traction forces are defined to remain constant. 3 u ( x,y, n ) cons tan t u ( x, m,z ) cons tant 2 (14) With the applied boundary displacement constraints, the average transverse shear strain is calculated as: 1 u2 23 dv V V 23 2 (15) n Equating the internal strain energy with the work done on the unit cell:
27 1 1 23 23 V 4 T23 2m 2n (16) 2 2 By substituting eq. (15) into eq. (16), it is proven that the average transverse shear stress is equal to the applied shearing stress: 23 T 23 (17) Therefore, the transverse shear modulus is calculated as follows: G 23 = T23 n U 2 (18) Validation of Predicted Properties A carbon-epoxy composite lamina is chosen to validate the unit cell model because of the availability of test data. The independent AS4 carbon fiber and 3501-6 epoxy material properties used in the analysis are displayed in Table 1. Table 2. Mechanical Properties of Fiber and Matrix Materials Ref. [23] E 1 (GPa) E 2 (GPa) v 12 v 23 G 12 (GPa) AS4 Carbon 235 14 0.20 0.25 28 3501-6 Epoxy 4.8 4.8 0.34 0.34 1.8 The elastic lamina properties obtained from the finite element model are compared to analytical and experimental results found in the literature. Hashin [1] and Whitney and Riley [43] attained results using an analytical approach based on energy principals. Sun [42] and Chamis [10] formulated an approach based on micromechanics which utilized displacement constraints and force equilibrium conditions. Meanwhile, Daniel and Lee [45] and Sun and Zhou [44] performed an experimental investigation to estimate the
28 effective elastic composite properties. Sun and Vaidya [33] validated the boundary displacement constraints applied to the unit cell model by building a RVE model consisting of multiple fibers to examine the deformed shape of a centrally located cell. Following the approach formulated by Sun and Vaidya, the modeling tools coupled with ABAQUS generated results that showed substantial agreement with analytical and experimental results found in the literature (Table 3). The out-of-plane lamina properties extracted from the FEA are further compared to results obtained using the self-consistent Field method (eq. 19). The property comparison between the FEA and analytical equations are presented in Table 4. While the analytical equations yield in-plane properties that agree quite well, the predicted out-of-plane properties differ substantially from both the FEA results and previous validation results. The deviation is introduced by the out-of-plane lamina bulk modulus (K 2 ) term which propagates through the remaining out-of-plane properties. This comparison is performed to illustrate that the two analysis methods are generating considerably different results.
29 Table 3. Mechanical Properties of Carbon/Epoxy Composite Lamina (Fv=0.6) Elastic Property (GPa) FEA Ref. [33] Ref. [1] Ref. [43] Ref. [42] Ref. [10] *Ref. [45] *Ref. [44] E 1L 142.7 142.6 142.9 142.9 142.6 142.9 142.0 139.0 E 2L 9.61 9.60 9.40 9.78 9.20 9.79 10.30 9.85 G 23L 3.09 3.10 3.42 - - 3.01 3.80 - v 12L 0.25 0.25 0.25 0.25 0.26 0.26-0.30 v 23L 0.35 0.35 0.39 - - 0.42 - - * experimental data (19) Table 4. Comparison of Lamina Properties Obtained from FEA and Self-Consistent Field Method Method E 1L E 2L v 12L v 23L G 23L FEA 142.7 9.61 0.25 0.35 3.09 Analytical 142.9 9.10 0.25 0.38 3.29
30 Investigation of Poisson s Effect Some researchers have assumed that cell boundaries on which no normal load acts are forced to remain stationary as part of the larger lamina and have chosen to neglect Poisson s effects (Figure 14). (a) Including Poisson s effect (b) Neglecting Poisson s effect Figure 14. Comparison of Deformed Shape with and without Poisson s Effect Therefore, the code is formulated with an option to include or neglect Poisson s effect allowing for easy comparison with analytical or experimental results. The omission of Poisson s effect yielded a substantial increase in the transverse tensile modulus of the lamina (Table 5) and cause for a significant reduction in the maximum effective stress developed in the fiber (Figure 15). This behavior occurs because there is more volume to resist the applied normal stress. Including Poisson s effect yield lamina properties much closer to those found in the literature (Table 3); therefore, Poisson s effect is included in all further analyses.
31 Table 5. Comparison of Lamina Properties Relative to Boundary Conditions (GPa) Boundary Conditions E 1L (GPa E 2L v 12L v 23L G 23L Include Poisson's Effect 142.7 9.61 0.25 0.35 3.089 Neglect Poisson's Effect 144.6 11.05 0 0 3.089 Percent Increase 1.3% 15.0% - - 0.0% (a) Included Poisson s effect max stress = 1,496 GPa (b) Neglected Poisson s effect max stress = 962 GPa Figre 15. Comparison of von Mises Stress with Varying Boundary Conditions Single Variant Sensitivity Analysis Core Methodology Typically in aircraft design and analysis, the effects of out-of-plane shear modulus, G 23, and Poisson s ratio, 23, on structural behavior are omitted, or these properties are roughly estimated. Understanding the sensitivity of structural performance results relative to these two properties will lead to effective test plans and improved structural analysis. A high sensitivity indicates that it is worth the time and resources for thorough material testing to obtain out-of-plane properties for each specific composite. If
32 it is determined that structural behavior is not sensitive to these out-of-plane properties, then designers and analysts can continue using current assumptions and methods with confidence. Behaviors that may be sensitive to out-of-plane properties include bearing, interlaminar stresses due to temperature loading, crack propagation, buckling, and stress distributions around holes. These complex and less understood material interactions are why out-of-plane composite properties (E 2L, v 23L, and G 23L ) are chosen for the focus of this sensitivity study. A single variant sensitivity analysis is performed according to the one-factor-at-atime (OFAT) methodology to examine the effect of constituent properties on out-of-plane lamina properties. One constituent parameter is varied while all others are held at a baseline value in order to monitor the influence of the single variant. The parameter is returned to a baseline value before the next parameter is subject to variation. An extensive overview of sensitivity analysis methods including OFAT is presented by Hamby [46]. Performing a sensitivity analysis through FEA can be considered timeconsuming, but the automation capabilities of the developed computational tools eliminate all user-intensive tasks associated with FEA of fibrous composites. A carbonepoxy composite material is once again used to demonstrate the usefulness and efficiency of the automated modeling framework. Determination of Input Parameter Range An extensive literature search for the material properties of AS4 carbon and 3501-6 epoxy resin revealed that available experimental test data is limited and inconsistent. The sparseness and questionable validity of the available constituent data makes carbon-
33 epoxy an exemplary material for investigation. The mean and standard deviation are used to establish a representative range for the two matrix properties (E m, v m ), five independent fiber properties (E 1f, E 2f, v 12f, v 23f, G 12f,), and the fiber volume fraction (V F ). The lower and upper bounds for each parameter are calculated as three standard deviations on either side of the calculated mean. The data collection and calculated bounds are presented in Tables 6 and 7. The collected data revealed no variability in both the in and out-of-plane Poisson s ratio of the carbon fiber; therefore, a lower and upper bound are approximated. Meanwhile, the data collected for the in-plane shear modulus of the fiber (G 12f ) followed a bimodal distribution. Therefore, an equivalent mean and standard deviation are calculated to span three deviations above and below the two observed means. These equivalent values are subsequently used in the input file generation code to establish a range for G 12f. The fiber volume fraction for the carbon-epoxy is selected to range from 0.5 to 0.7 as this is the range most often mentioned in the literature [3]. The mean values for all input properties are used to establish a baseline set of elastic lamina properties. Next, six FE simulations are performed over each input property range according to the OTAF methodology to evaluate the sensitivity of each out-of-plane lamina property. Results and Discussion The sensitivity results of the three out-of-plane lamina properties (E 2L, v 23L, G 23L ) relative to the eight independent constituent properties (E m, v m, E 1f, E 2f, v 12f, v 23f, G 12f, F v ) are displayed on the following pages in Figures 16-18. The relative influence of each input property is readily observed through the slope of each graph.
34 Table 6. 3501-6 Epoxy Resin Properties from Literature Search Reference E m (GPa) v m [47] 4.2 0.34 [26] 4.3 0.35 [48] 4.3 0.34 [24] 4.4 0.34 [49] 4.2 0.34 [50] 4.4 0.34 [51] 4.8 0.34 [52] 4.2 - [3] 4.3 0.35 Mean 4.349 0.343 Calculated s 0.184 0.005 Lower Bound 3.796 0.329 Upper Bound 4.902 0.356 Table 7. AS4 Carbon Fiber Properties from Literature Search Reference E 1f (GPa) E 2f (GPa) v 12f v 23f *G 12f (GPa) [53] 224.0 14.0 0.20 0.25 14.0 [10] 213.7 13.8 0.20 0.25 13.8 [47] 225.0 15.0 0.20-15.0 [54] 234.0 14.0 - - - [55] 228.0 - - - - [56] 231.0 - - - - [24] 221.0 13.8 0.20 0.25 13.8 [3] 235.0 15.0 0.20-27.0 [57] 231.0 - - - - [23] 235.0 14.0 0.20 0.25 28.0 [58] 235.0 - - - - [48] 235.0 14.0 0.20-28.0 Mean 229.0 14.20 0.200 0.250 14.15 (27.67) Calculated s 6.86 0.501 0.000 0.000 0.574 (0.577) Effective s 6.86 0.501 0.033 0.033 2.829 Lower Bound 208.4 12.70 0.100 0.150 12.43 Upper Bound 249.5 15.70 0.300 0.350 29.40 *Bimodal distribution was observed. Equivalent values: x = 20.913 GPa, s = 2.829
E 2L (GPa) E 2L (GPa) E 2L (GPa) E 2L (GPa) E 2L (GPa) E 2L (GPa) E 2L (GPa) E 2L (GPa) 35 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 E m (GPa) 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 v m 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 205 215 225 235 245 255 E 1f (GPa) 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 12.5 13 13.5 14 14.5 15 15.5 16 E 2f (GPa) 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 0.07 0.12 0.17 0.22 0.27 0.32 v 12f 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 0.13 0.18 0.23 0.28 0.33 0.38 v 23f 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 12 14 16 18 20 22 24 26 28 30 G 12f (GPa) 10.40 10.10 9.80 9.50 9.20 8.90 8.60 8.30 0.48 0.52 0.56 0.6 0.64 0.68 0.72 F v Figure 16. Sensitivity of E 2L
36 v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 E m (GPa) v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 v m v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 205 215 225 235 245 255 E 1f (GPa) v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 12.5 13 13.5 14 14.5 15 15.5 16 E 2f (GPa) v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 0.07 0.12 0.17 0.22 0.27 0.32 v 12f v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 0.13 0.18 0.23 0.28 0.33 0.38 v 23f v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 12 14 16 18 20 22 24 26 28 30 G 12f (GPa) v 23L 0.420 0.400 0.380 0.360 0.340 0.320 0.300 0.48 0.52 0.56 0.6 0.64 0.68 0.72 F v Figure 17. Sensitivity of v 23L
G 23L (GPa) G 23L (GPa) G 23L (GPa) G 23L (GPa) G 23L (GPa) G 23L (GPa) G 23L (GPa) G 23L (GPa) 37 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 E m (GPa) 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 v m 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 205 215 225 235 245 255 E 1f (GPa) 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 12.5 13 13.5 14 14.5 15 15.5 16 E 2f (GPa) 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 0.08 0.12 0.16 0.2 0.24 0.28 0.32 v 12f 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 0.14 0.18 0.22 0.26 0.3 0.34 0.38 v 23f 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 12 14 16 18 20 22 24 26 28 30 G 12f (GPa) 3.45 3.35 3.25 3.15 3.05 2.95 2.85 2.75 2.65 2.55 0.48 0.52 0.56 0.6 0.64 0.68 0.72 F v Figure 18. Sensitivity of G 23L
38 The linear relation of the sensitivities allows for the sensitivity data to be condensed into three effective sensitivity diagrams (Figures 19-21), yet a brief explanation may be necessary to fully understand each diagram. The centerline of the bar chart represents the baseline case in which the mean values for all input properties were used. Solid bars designate the decrease in the considered lamina property caused by the minimum value of the input parameter, while hatched bars designate the increase in the lamina property caused by the maximum value of the input parameter. The associated data labels were calculated as the percent change of the lamina property relative to the baseline value. For example, the data labels characterizing the influence of matrix modulus (E m ) on the transverse tensile modulus of the lamina (E 2L ) were calculated as such. % E E (min) E (base ) 2L 2L 2L E2L(base ) 100 % E E (max) E ( base ) 2L 2L 2L E2L( base ) 100 The fiber volume ratio is clearly observed to have the strongest impact on the outof-plane lamina properties. An influence rank is assigned to each remaining parameter by averaging its influence on all three out-of-plane lamina properties. The out-of-plane Poisson s ratio of the fiber (v 23f ) ranks second followed by the matrix modulus (E m ), matrix Poisson s ratio (v m ), and the out-of-plane fiber modulus (E 2f ).
39 Em -6.1% 5.4% vm -2.4% 3.0% E1f 0.0% 0.0% E2f -6.0% 5.6% v12f v23f G12f -0.1% -0.8% 0.0% 0.1% 0.9% 0.0% Fv -10.6% 11.1% 7.8 8.1 8.4 8.7 9 9.3 9.6 9.9 10.2 10.5 10.8 E 2L (GPa) Figure 19. Sensitivity of E 2L Relative to Constituent Properties Em -0.9% 0.7% vm -9.1% 10.4% E1f E2f v12f -0.1% -0.8% -0.4% 0.1% 0.8% 0.4% v23f -13.8% 13.7% G12f 0.0% 0.0% Fv -7.1% 8.3% 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 v 23L Figure 20. Sensitivity of v 23L Relative to Constituent Properties Em -8.7% 8.0% vm E1f E2f v12f v23f G12f -1.3% 0.0% -3.9% 0.0% -2.6% 0.0% 1.4% 0.0% 3.4% 0.0% 2.7% 0.0% Fv -11.3% 16.1% 2.46 2.56 2.66 2.76 2.86 2.96 3.06 3.16 3.26 3.36 3.46 G 23L (GPa) Figure 21. Sensitivity of G 23L Relative to Constituent Properties
40 The established range for each input caused considerable influence on the out-ofplane lamina properties; however, with this high fidelity analysis, it cannot be determined if the variation in the out-of-plane lamina properties is significant enough to effect overall part performance. This unit cell modeling framework acts to bridge between constituent level and lamina level properties, yet the framework must further be coupled with larger scale analyses to evaluate the influence on structural performance. It is important to note that a total of 96 simulations are performed to gather the effective out-of-plane lamina properties, yet the total simulation time amounts to less than 29 minutes (~18 seconds per simulation). The simulations require limited computing resources, and an average laptop computer is able to perform other tasks during simulation with no noticeable slow down. Conclusion An automated FE based approach is developed and used to evaluate the sensitivity of out-of-plane composite lamina properties relative to constituent material properties. The modeling approach is shown to strongly agree with prior analysis results and proves to be a feasible and effective means of studying composite lamina properties. The developed unit cell modeling tools are the first in a line of multi-scale modeling tools intended to study the effect material uncertainties across multiple scales.
41 Chapter 4 Sensitivity Analysis of Composite Lamina with Multiple Fiber Materials Introduction Highly customized fiber-reinforced composites exhibit promise as an adaptable, high-performing material that can be used across many industries. For example, an advanced FRC material may contain different types of reinforcing fibers such that the properties supplied by each fiber may be used to achieve an advanced engineering function [59]. To create optimized parts designed from these highly customized materials, engineers need the capability to rapidly analyze many constituent combinations and material configurations. Leveraging today s available computing resources by performing composite analysis through simulation provides a cost-effective approach to achieving this goal. An encompassing simulation environment containing all relevant constituent information such as cost, weight, mechanical properties, and pre-made FE models will allow engineers to economically explore material combinations, develop effective testing programs, and ultimately design optimized parts that satisfy multiple engineering functions. This chapter outlines the development and application of computational modeling tools intended to help build this encompassing simulation environment. The modeling tools developed for this lamina study are similar to those discussed in the previous chapter of this thesis. The same foundation is used to develop the load schemes, boundary displacement constraints, and homogenization routine in order to obtain the
42 homogenized lamina properties. However, the model to be generated by the input files consists of lamina RVE comprised of multiple unit cells aligned for a square packing sequence (Figure 22). This allows the user to specify different material properties for each cell within the RVE. Code written in C++ and batch scripting are once again used to generate input files and automate the FEA process. The modeling tools are used to evaluate the out-of-plane mechanical properties of laminas consisting of variable percentages of two different fiber materials. The framework is then used to identify a combination of the two fiber types that achieves optimal out-of-plane stiffness, deflection, and weight. Figure 22. Example RVE of a Lamina
43 Finite Element Model Model Structure The modeling tools generate input files containing the load conditions (longitudinal and transverse tension or shear) and periodic boundary conditions in order to obtain effective lamina properties through FEA in ABAQUS. However, additional features allow the user to specify the lamina size (n rows by m columns) and material properties for each fiber in the lamina, allowing for multiple fiber types to be modeled within a single lamina RVE. As discussed in the previous chapter, the input file generation code allows the user to control mesh density, fiber volume fraction, and the applied boundary conditions (inclusion or omission of Poisson s effect). After simulations are executed, the previously outlined extraction code retrieves the boundary displacements from each model output file, calculates the homogenized properties, and deposits the results into a single CSV file. Validation The lamina model used for validation consists of a three by three periodic arrangement of square packed fibers. The lamina is constructed with AS4 carbon fiber and 3501-6 Epoxy each with the average properties obtained from the literature search (Table 6-7). The out-of-plane homogenized properties acquired from the lamina model are compared to the homogenized properties obtained from the unit cell model consisting of identical constituent properties. Figure 23 compares the von Mises effective stress between the two models.
44 (a) Lamina in transverse tension (b) Unit cell in transverse tension (c) Lamina in transverse shear (d) Unit cell in transverse shear Figure 23. Comparison of Deformed Shape and Effective von Mises Stress Contours Obtained from the Lamina and Unit Cell Models
45 The effective stress contours, maximum and minimum effective von Mises stress, and homogenized properties are shown to agree between the models (Table 8). Table 8. Comparison of Effective Stress and Properties Obtained from the Lamina and Unit Cell Models (AS4 Carbon / 3501-6 Epoxy) Model Max. Stress (Tension/Shear) Min. Stress (Tension/Shear) E 2L (GPa) v 23L G 23L (GPa) Unit Cell 1,496 / 2,165 369 / 1,007 9.305 0.353 2.928 Lamina 1,496 / 2,166 369 / 1,006 9.305 0.353 2.927 Sensitivity Analysis of Composite Lamina Consisting of Multiple Fiber Types Designing composite lamina from multiple fiber types enables an engineer to take advantage of the favorable properties possessed by each fiber material. Therefore, the modeling framework is expanded to investigate the behavior of composite lamina consisting of two different fiber materials. The two fiber materials chosen for the analysis were AS4 carbon and Kevlar-49 Aramid fiber. AS4 carbon fiber has exceptional strength and stiffness in both compression and tension, but exhibits little toughness relative to other fiber materials. Kevlar-49 fiber is known for impact resistance because of its high tensile strength and toughness and is lighter (~20%) and cheaper than AS4 carbon. However, Kevlar-49 has a relatively low compressive strength, which limits its structural applications. Therefore, combining the two fiber materials in single composite lamina could provide for a cost-effective, tough material that retains a high degree of multi-directional strength and stiffness. The average properties of the two fiber materials
46 are displayed in Table 9. It is important to note that AS4 carbon is substantially more resilient to deformation than Kevlar-49 for in and out-of-plane loading. Table 9. Avg. Properties of AS4 Carbon and Kevlar-49 from Literature Search (GPa) Fiber Material E 1f E 2f v 12f v 23f G 12f G 23f AS4 Carbon 229.0 14.20 0.200 0.250 20.91 5.680 Kevlar-49 127.1 3.993 0.523 0.275 14.34 1.251 Fiber Percentage Study Using the validated modeling framework, a sensitivity analyses is performed to examine the behavior of a three by three lamina consisting of an increasing percentage of Kevlar-49 fibers. Nine different composite lamina models are used to establish a spectrum of fiber combinations beginning with a lamina comprised of all AS4 carbon fibers and ending with a lamina comprised of all Kevlar-49 fibers (Figure 24). The modeling framework is then used to generate the input files for each lamina model, run the simulation in ABAQUS, and output the out-of-plane lamina properties into a single CSV file. This demonstration study focuses only on the out-of-plane properties, although the modeling framework is not limited to out-of-plane behavior. The lamina property calculations adhere to the formulation presented by Sun and Vaidya [33] and are calculated as follows (eq. 20). E2 L 22 22 P2 L 2u 2 u v23 L u 3 2 G 23L 23 23L T u 23 2 L u 3 (20)
47 Figure 24. Lamina Combinations for Sensitivity Analysis (Black: Kevlar-49) The terms P 2 and T 23 are the traction loads applied to the lamina model, u 2 and u 3 are the boundary displacements obtained from the FEA, and L is the length of the lamina side. The results from the sensitivity study are displayed in Figures 25 and 26. It is observed that the location of the stiffer fibers have a fairly substantial influence on the out-of-plane Poisson s ratio of the lamina (v 23L ) indicated by the jump in the graph located between configuration 4 and 5. The jump in v 23L is due to the centrally located AS4 carbon fiber that is present in lamina combination 4, but not present in lamina combination 5. Although this is not intuitive or obvious, the AS4 carbon fibers acting in series limit the relative vertical contraction (u 3 ) in the lamina more profoundly than the horizontal extension (u 2 ). The automation and extraction tools allow the user to identify optimal material combinations. The usefulness is demonstrated through a simple design optimization problem, where the goal is to identify a material combination that weighs less and possesses higher toughness than a traditional carbon-epoxy composite, but retains a high degree of stiffness. Just by examining the previously generated sensitivity results it is
48 observed that the AS4 carbon/kevlar-49 combination just above 40% Kevlar-49 provides the maximum transverse modulus (E 2L ) and minimal transverse Poisson s ratio (v 23L ). Figure 27 displays a normalized plot of the transverse modulus compared to the total composite weight, and from this figure it is observed that the combination to maximize E 2L and minimize weight also occurs just above 40% Kevlar-49. Therefore, it is determined that a combination consisting of 4 Kevlar-49 fibers to every 5 AS4 carbon fibers accomplishes the design goals. From a pure volumetric consideration, this combination should culminate in a 20% increase in composite lamina toughness. While manufacturing a composite material with an exact fiber configuration is unrealistic, manufacturing materials with reasonable degree of accuracy in fiber composition is feasible.
Relative Weight G 23L (GPa) E 2L (GPa) 49 10.0 8.0 6.0 4.0 0.400 0.384 0.368 0.352 v 23L 2.0 E2L v23l 0.336 0.0 0.320 0% 20% 40% 60% 80% 100% Percent Kevlar-49 Fibers Figure 25. Effect of Increasing Kevlar-49 Percentage on E 2L and v 23L 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0% 20% 40% 60% 80% 100% Percent Kevlar-49 Fibers Figure 26. Effect of Increasing Kevlar-49 Percentage on G 23L Relative E 2L 1.0 0.0 0.8 0.2 0.6 E2L 0.4 0.4 weight 0.6 0.2 0.8 0.0 1.0 0% 20% 40% 60% Percent Kevlar-49 Fibers 80% 100% Figure 27. Bi-Fiber Lamina Optimization Problem
50 Fiber Configuration Study In this section, configuration refers to the location of the different fiber materials within the lamina RVE. An analysis was performed to examine the effect of fiber configuration on the out-of-plane lamina properties because the location of the stiffer fibers appeared to cause considerable influence on the out-of-plane lamina properties (E 2L and v 23L ). Several lamina models were configured consisting of exactly four AS4 carbon and five Kevlar-49 fibers randomly located throughout a lamina RVE (Figure 28). The influence of configuration on the out-of-plane lamina properties is displayed in Figure 29. It was observed that configuration alone played a minimal role in the effective out-of-plane lamina properties. The percentage difference between the maximum and minimum achieved E 2L was calculated to be just 4.2%, while the percentage difference between the maximum and minimum v 23L was calculated to be 13.8%. Figure 28. Lamina Configurations for Sensitivity Analysis (Black: Kevlar-49)